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Let $T$ be a tournament with nondecreasing score sequence $R$ and $A$ be itstournament matrix. An upset of $T$ corresponds to an entry above the maindiagonal of $A$. Given a feasible score sequence $R$, Fulkerson~(1965) gave asimple recursive construction for a tournament with score sequence $R$ and theminimum number of upsets, and Hacioglu et al. (2019) provided a constructionfor all of such tournament matrices. Let $U_{\min}(R)$ denote the set oftournament matrices with score sequence $R$ that have minimum number of upsets.Brauldi and Li~(1983) characterized the strong score sequences $R$ ($R$ isstrong if a tournament $T$ with score sequence $R$ is strongly connected) with$|U_{\min}(R)|=1$. In this article, we characterize all feasible scoresequences $R$ with $|U_{\min}(R)|=1$ and give an explicit formula for thenumber of the feasible score sequences $R$ with $|U_{\min}(R)|=1$.

The score sequences with unique tournament that has minimum number of upsets